Asymptotics of the expected posterior

Let (X₁, . . . , X_n) be independently and identically distributed observations from an exponential family p_θ equipped with a smooth prior density w on a real d-dimensional parameter θ. We give conditions under which the expected value of the posterior density evaluated at the true value of the parameter, θ₀, admits an asymptotic expansion in terms of the Fisher information I(θ₀), the prior w, and their first two derivatives. The leading term of the expansion is of the form n^d/2C₁(d, θ₀) and the second order term is of the form n^d/2-1c₂(d, θ₀, w), with an error term that is o(n^d/2-1). We identify the functions C₁ and C₂ explicitly. A modification of the proof of this expansion gives an analogous result for the expectation of the square of the posterior evaluated at θ₀. As a consequence we can give a confidence band for the expected posterior, and we suggest a frequentist refinement for Bayesian testing.